1. Technical Field of the Invention
The present invention relates to using Global Positioning System (GPS) signals received by nonaligned antennas mounted on a vehicle to determine the attitude of the vehicle, and, in particular, to correcting measurements of the GPS signals received by the nonaligned antennas to compensate for the respective orientations of the nonaligned antennas.
This application is based on "Development and Flight Demonstration of a GPS Receiver for Space", a dissertation submitted to the Department of Aeronautics and Astronautics and the Committee on Graduate Studies of Stanford University in partial fulfillment of the requirements for the degree of Doctor of Philosophy, by Edgar Glenn Lightsey, February, 1997, UMI Number 9723388, UMI, A Bell & Howell Information Company, 300 North Zeeb Road, Ann Arbor, Mich. 48106-1346 USA, 313/761-4700, and which is incorporated herein by reference, and on GPS Based Attitude Determination On Nonaligned Antenna Arrays, by E. Glenn Lightsey, NASA Goddard Space Flight Center, and Bradford W. Parkinson, Ph.D., Stanford University, technical paper, Institute of Navigation GPS-96 Conference, incorporated herein by reference.
2. Background Art
The principles of attitude determination using GPS carrier phase interferometry are well documented in the literature (C. E. Cohen, Attitude Determination Using GPS, Ph.D. Dissertation, Stanford University, December 1992 and C. E. Cohen, "Attitude Determination," Global Positioning System: Theory and Applications, vol. II, American Institute of Aeronautics and Astronautics (AIAA), 1996), pps. 519-537, incorporated herein by reference.
The use of the Global Positioning System (GPS) for vehicle attitude determination is disclosed in U.S. Pat. No. 5,548,293, "System and method for generating attitude determinations using GPS", issued Aug. 20, 1996, to Clark E. Cohen, incorporated herein by reference.
A method for comparing the relative phase of carrier signals received from Global Positioning System satellites to determine the roll, pitch, and azimuth attitude of ships, aircraft, land vehicles, or survey instruments using a maximum likelihood estimation (MLE) optimum solution over a full range of integers and vehicle attitudes is discussed in "Method and Apparatus for Maximum Likelihood Estimation Direct Integer Search in Differential Carrier Phase Attitude Determination Systems", U.S. Pat. No. 5,296,861 to Donald T. Knight, incorporated herein by reference.
The spherical trigonometry of phase angle measurements is disclosed in Spacecraft Attitude Determination and Control, J. R. Wertz, ed., D. Reidel Co., 1978.
Vehicle attitude determination using GPS signals is made, taking into account the differential phase received by, for example, three antennas from the same GPS signals. GPS signal acquisition is described in "Development and Flight Demonstration of a GPS Receiver for Space", id.
Conventional methods of vehicle attitude determination include using the differential phase of the GPS signals received by several different antennas, which are all aligned with each other and pointed in the same direction, relative to the GPS signal being received. In determining the attitude of a vehicle using the differential phase method, an absolute phase of the GPS signal received by one antenna is subtracted from the absolute phase of the GPS signal received by a second antenna, as shown below: EQU .DELTA..phi.=.phi..sub.2 -.phi..sub.1
The above-mentioned equation provides information about how far apart a first antenna is from a second antenna along the GPS signal line of sight. By calculating the differential phase (.DELTA..phi.), common error sources interjected by the GPS signal and by the reception of the GPS signal by the antennas are removed. For example, the GPS signal is a right handed circular polarized (RHCP) signal, and the phase contribution due to this polarization effect is removed by calculating the differential phase of the GPS signal.
Signals broadcast from Global Positioning System (GPS) satellites have previously been used to determine vehicle position. Attitude determination of a platform or vehicle using carrier signals has not traditionally been considered a standard GPS measurement, but attitude determination of a vehicle using GPS carrier signals greatly enhances the overall utility of a device sensing GPS signals, and has been performed in recent years. All of the information required for vehicle attitude control, and, further, for autonomous position control and precise timing, is available in a single package at reasonable mass, power, and cost. Accordingly, this consolidation of sensory resources makes the GPS receiver a powerful component for many Low Earth Orbit (LEO) spacecraft applications.
The concept of attitude determination using GPS carrier phase interferometry was first demonstrated by Greenspan, et al, in 1982 (R. L. Greenspan, A. Y. Ng., J. M. Przyjemski, J D. Veales, "Accuracy of Relative Positioning by Interferometry with Reconstructed Carrier GPS: Experimental Results", Proc. Of the Third Intel. Geodetic Symposium on Satellite Positioning, Las Cruces, N. M., Mar. 1982). After several initial demonstrations on platforms, ships, and aircraft during the 1980s (see L. R. Kruczynski, P. C. Li, A. G. Evans, B. R. Hermann, "Using GPS to Determine Vehicle Attitude: USS Yorktown Test Results," Proc. Int. Tech. Mtg., Institute of Navigation (ION), Colorado Springs, Colo., September 1989, and F. van Graas, M. Braasch, "GPS Interferometric Attitude and Heading Determination: Initial Flight Test Results," Navigation, vol. 38, Fall, 1991), the first commercial terrestrial receivers capable of performing 3-axis (x, y, and z) GPS carrier phase-based attitude determination were marketed in the early 1990s. Several algorithmic improvements to the state of the art were made and the accuracy of the attitude determination system was demonstrated through aircraft experiments in 1992 (C. E. Cohen, Attitude Determination Using GPS, Ph.D. Dissertation, Stanford University, December 1992).
In the present application, "zenith" refers to a direction directly away from the center of the Earth (i.e., "up"), "nadir" refers to a direction directly into the center of the Earth (i.e., "down"). In addition, a master antenna is one from which relative distances and measurements are calculated, and slave antennas are those antennas for which relative distances to and measurements by the master are calculated.
A widely used method of GPS attitude determination is illustrated in FIG. 1, and is referred to as the Carrier Phase Interferometry Method of Attitude Determination. As shown in FIG. 1, two antennas 10 are separated by a known fixed distance, [b].sub.B (also referred to as the antenna baseline vector), expressed in a body referenced coordinate frame `B`. Carrier wave signals 11, such as L1 carrier signals (wavelength .lambda..congruent.19 cm), originate from a GPS satellite 12 along a known line of sight [s].sub.E, expressed as a unit vector in an external reference frame `E`. The known line of sight implies that the position of the GPS satellite 12 and the position of the receiver on which the antennas 10 are placed are at least approximately known. Because the GPS satellite 12 is far away from the antennas 10, the carrier wavefronts 14 are essentially planar.
The range projection of the antenna baseline onto the line of sight vector may be expressed as: EQU .DELTA.r=s.sup.T b (1)
.DELTA.r can be expressed in any units of distance, such as meters, but is more conveniently represented in wavelengths (.lambda.) for this derivation. The superscript T indicates the transpose of the matrix. Recognizing that one carrier phase wavefront is indistinguishable from the next, the true range projection can be expressed in terms of an ideal differential carrier phase measurement: EQU .DELTA.r=s.sup.T b=.DELTA..phi.-k (2)
where .DELTA..phi. is the differential carrier phase measurement (a fractional number between 0 and 1 wavelength) and k is a differential integer that accounts for the integer number of wavelengths in the distance .DELTA.r. For example, if .DELTA.r&lt;19 cm, k=0 regardless of the orientation of b. The minus sign on k results from construction in FIG. 1.
.DELTA..phi. can in general be outside the bounds of one wavelength due to phase wraparound. The integer k is adjusted only at discrete times (such as the first attitude computation), and during other times the integer k is treated as a constant. .DELTA..phi. may therefore contain an integer component as well, after the carrier phase measurement has gone through more than one rotation of 2.pi.; .DELTA..phi. is nonetheless termed the `fractional` carrier phase measurement to distinguish .DELTA..phi. from the integer term k.
Since the baseline vector b is known in the vehicle body reference frame, and the GPS signal line of sight s is known in the external reference frame, the attitude of the body frame may be expressed with respect to an ideal differential carrier phase measurement as: EQU .DELTA.r=s.sup.T b=[s.sup.T ].sub.E A.sub.E&gt;B.sup.T [b].sub.B =.DELTA..phi.-k (3)
where A.sub.E&gt;B represents the 3.times.3 direction cosine transformation matrix from the external reference to the body reference. If the differential integers ("cycle ambiguities") are known, this represents a nonlinear equation that can be solved to obtain the attitude of the antenna platform 10 in the external reference frame. A.sub.E&gt;B has three independent unknowns, meaning that this equation may be theoretically solved if there are as few as three differential phase measurements, for example from three antennas and two GPS signals, provided these measurements span a three dimensional space. However, two GPS signals along the same line of sight would not span a three dimensional space.
Equation 3 is modified to account for real measurements by including two additional terms: EQU [s.sup.T ].sub.E A.sub.E&gt;B.sup.T [b].sub.B =.DELTA..phi.-k-.beta.-v(4)
.beta. is a known calibration constant which is a function of the receiver hardware and the antenna pair in question. .beta. is called the line bias, and represents the electrical line length from the antenna phase center through the cable to the measurement point inside the receiver. In this difference equation, the term line bias actually refers to the differential bias; i.e., the line bias of the slave antenna electrical path minus the master antenna electrical path. The line bias quantity is a function of the GPS antenna/receiver hardware and can be determined in advance and saved using a calibration technique (C. E. Cohen, Attitude Determination Using GPS, Ph.D. Dissertation, Stanford University, December 1992).
The remaining term, v, is an additive measurement noise term, and includes all noise effects such as multipath, carrier tracking noise, bias drifts, and others. Once again, what is measured in this case is a differential noise between the slave and master antenna; only non-common mode noise sources remain after the differencing operation. Clock offset, for example, is not a differential noise source, since it is common to each antenna. The noise term v has time correlated properties that in some cases may be modeled or calibrated, but in this analysis these effects are treated as measurement error.
Equation 4 is the fundamental GPS differential carrier phase measurement equation. It may be expanded to account for all GPS satellites being tracked (i), across all antenna baselines (j), to accommodate all measurements made during one sample interval: EQU .DELTA..phi..sub.ij =[s.sub.i.sup.T ].sub.E A.sub.E&gt;B.sup.T [b.sub.j ].sub.B +k.sub.ij +.beta..sub.j V.sub.ij (5)
Equation (5), then, is the basic vehicle attitude determination equation. In the above-mentioned equation, .DELTA..phi..sub.ij is the measured differential phase, k.sub.ij is the integer cycle ambiguity, .beta..sub.j is the electrical line bias, and V.sub.ij is the noise. The equation is rearranged so that the fractional carrier phase measurement appears on the left and all other terms are separated on the right. With enough measurements, it is possible to solve for the cycle ambiguities, k.sub.ij, and the direction cosine matrix A.sub.E&gt;B. Although, as previously noted, the attitude may be theoretically determined from as few as three ideal (i.e., noiseless) differential carrier phase measurements, more measurements are required in practice to account for the presence of the measurement noise.
Once the cycle ambiguities k.sub.ij are known, the cycle ambiguities may be removed from the problem of determining the attitude of the vehicle, and if A.sub.E&gt;B is approximately known as (A.sub.E&gt;B).sub.0, a perturbation equation may be developed as follows: EQU (.DELTA..phi..sub.ij).sub.0 =[s.sub.i.sup.T ].sub.E (A.sub.E&gt;B.sup.T).sub.0 [b.sub.j ].sub.B +k.sub.ij +.beta..sub.j (6) EQU .delta..phi..sub.ij =.DELTA..phi..sub.ij -(.DELTA..phi..sub.ij).sub.0 =[s.sub.i.sup.T ].sub.E A.sub.E&gt;B.sup.T [b.sub.j ].sub.B -[s.sub.i.sup.T ].sub.E (A.sub.E&gt;B.sup.T).sub.0 [b.sub.j ].sub.B +v.sub.ij(7)
For sufficiently small perturbations, the attitude matrix A.sub.E&gt;B may be linearized for small rotations about the body axes: ##EQU1## Then ##EQU2## The superscript X indicates cross matrix of a vector. Equation 10 represents a linearized sensitivity equation between the measured differential carrier phase and the perturbation to the initial attitude guess. h.sub.ij is a 1.times.3 row vector that may be thought of as a set of linearized sensitivity coefficients for every measurement equation. For a given sample, all `valid` differential phase measurements (i.e., those for which the cycle ambiguities are known) are stacked into a single linearized vector equation: ##EQU3##
This equation presents an overdetermined linear system with additive noise whose solution may be obtained by minimizing the residual in the least squares sense (or according to any other appropriate performance index). The solution, .delta..phi., is a 3.times.1 vector correction of small rotations to the direction cosine matrix (A.sub.E&gt;B).sub.0 as defined by Equation 8. The solution may be obtained iteratively using the previous epoch solution as an initial guess.
A typical GPS signal receiver 13 of the prior art, used for navigation and attitude determination, is shown in FIG. 2. The GPS receiver 13 shown in FIG. 2 includes a navigation board 15 and an attitude board 17. As shown in FIG. 2, GPS carrier signals are received by antennas 10, the signal strength is enhanced by pre-amplifiers 19, and delivered to mux 21 through 50 .OMEGA. coaxial cables. The output of the mux 21 is presented to the RF section 23, then is delivered through 6 channels 25 to processor 27 (Motorola 68000). The foregoing aspects of the GPS receiver 13 are based on the Trimble Advanced Navigation Sensor (TANS) Vector, manufactured by Trimble Navigation, Ltd. Output from both processor 27 and processor 29 is then serially transmitted outside of the GPS receiver 13.
The flow of information for differential, carrier phase-based attitude determination is summarized in FIG. 3. Many types of measurements are combined in different ways to produce the ultimate output product, real-time attitude solutions 36. Prior to real-time operation, calibration measurements 16 are taken to determine the antenna line biases and baseline vectors in the body reference frame. This information is then saved for later use. During the real-time operation, the receiver position estimate is obtained from a position fix, if available, or an orbit propagator solution 22.
The GPS satellite position is computed from the broadcast GPS ephemeris 24 and combined with the receiver position 26 to produce the signal line of sight vector, [s].sub.E. This information is collected along with differential carrier phase measurements 20 which are used to resolve the carrier cycle ambiguities. The integer resolution procedure 28 is solved once by conventional methods, and then removed from the attitude determination problem as in Equations 6 and 7. Once the integers are obtained, these values are periodically updated as new measurements are added and real-time attitude solutions are computed in the method of Equations 7 to 13. As long as the sample rate is relatively fast compared to the vehicle dynamics, the previous sample may serve as an initial guess to the next solution. Integrity checks 30 are readily available in the form of the solution residual, and other parameters, during both the bootstrap integer resolution 28 and the real-time attitude solution processes 34.
Equations 6 and 7 assume that the line bias terms .beta..sub.j are known in advance and may be subtracted from the available measurements. In fact, these parameters may not be constants nor well known in advance. An example of a method for determining vehicle attitude is to bypass the need to know the line bias term by double-differencing the carrier phase measurements along the same baseline vector (R. Fuller, S. Gomez, L. Marradi, J. Rodden, "GPS Attitude Determination From Double Difference Differential Phase Measurements," ION GPS-96, Kansas City, Mo., September 1996): EQU .gradient..DELTA..phi..sub.12j =[(s.sub.1 -s.sub.2).sup.T ].sub.E A.sub.E&gt;B.sup.T [b.sub.j ].sub.B +(k.sub.ij -k.sub.2j)+(v.sub.1j -v.sub.2j)(14)
This formulation circumvents the problem of calibrating and/or modeling .beta..sub.j at the cost of a decrease in the number of available total measurements, which are reduced by the double differencing operation. The double-difference measurements are also more noisy than in the single-difference case.
The most common LEO spacecraft attitude determination applications are classified in terms of three types of pointing: Earth referenced, inertially referenced, and spinning. These may be contrasted with typical static terrestrial environments, such as survey applications, and dynamic terrestrial environments, such as aviation or maritime applications.
There are significant differences between terrestrial and space GPS carrier phase based attitude determination, such as motion and visibility, as shown in Table 1.
A substantive difference between terrestrial and LEO space applications concerns the magnitude and the cause of the motion in the carrier phase measurements as recorded by the GPS receiver, and their magnitudes relative to GPS line of sight motion. The cause of relative motion is important to the structure of cycle ambiguity resolution equations. Cycle ambiguity is discussed in Attitude Determination Using GPS, by C. E. Cohen, Ph.D. dissertation, Stanford University, December, 1992, in Development and Flight Demonstration of a GPS Receiver for Space, id, and herein below.
TABLE 1 ______________________________________ Comparison of Terrestrial and Orbital GPS Attitude Determination Environments Orbital Application Terrestrial Earth Inertially Type Static Dynamic Pointed Pointed Spinning ______________________________________ Antenna Zenith Zenith Full Sky Pointing GPS Line of 180 degrees per 180 degrees per Sight 360 minutes 45 minutes Motion Vehicle None Several &lt;1 rev. per 0.1- Dynamics deg. orbit = 20 + rev. Per &lt;4 deg. per per min. minute minute Carrier None More Same More Phase Motion Relative to GPS Line of Sight Motion ______________________________________
The other significant difference between terrestrial and space applications, which is addressed by the present invention as explained herein below, is signal visibility.
In terrestrial applications, whether dynamic or static, the antenna array is generally always pointed approximately upwards. The GPS Constellation resides within the hemisphere to which the antenna array points. Even in the most demanding applications, such as aircraft attitude determination during a steep bank, the bank angle rarely exceeds 45 degrees. However, in some terrestrial applications such as mobile communications, even though all of the antennas in the array may be pointed approximately upwards, all of the antennas may not be aligned with each other (pointed in the same direction).
For an inertially fixed or spinning spacecraft, however, there is no hemisphere on the vehicle body that points in the direction of the GPS constellation all the time. The foregoing pointing profiles are shown schematically in FIGS. 4(A), 4(B), and 4(C).
As shown in FIGS. 4(A), 4(B), and 4(C), arrows 38 indicate the direction that the GPS antennas 10 (not shown in FIGS. 4(A), 4(B), and 4(C)) are pointed. In addition, FIGS. 4(A), 4(B), and 4(C) show the relationship between the direction 38 in which the GPS antennas 10 are pointed and the GPS constellation visibility 39.
In FIG. 4(B), an inertially fixed axis that is pointed towards the zenith vector at one time will be aligned with the nadir vector a half orbit later. Even in the case of a nadir pointed spacecraft of FIG. 4(C), there may be transitional or contingency pointing modes where vehicle alignment is not fixed to one hemisphere.
Further, in the case of a vehicle in Low Earth Orbit (LEO), it is often advantageous to tilt the receiving antennas 10 with respect to each other to improve total coverage. In this case, the alignment of the antennas receiving GPS carrier signals must be taken into account when using GPS carrier signals to determine the attitude of the vehicle. However, prior art attitude determination systems using GPS carrier signals have not taken the alignment of the antennas into consideration in determining the attitude of the vehicle.
In "System and Method for Generating Attitude Determinations using GPS", U.S. Pat. No. 5,548,293 to Cohen, the use of a GPS receiver for determining the attitude of a moving vehicle in conjunction with four antennas mounted thereon is described. However, in the Cohen apparatus, all of the GPS antennas are pointed in the same direction, and therefore, look to the same part of the sky.
If the antennas, which exhibit coverage patterns of 180.degree. (i.e., the antennas receive signals from or see through an angle of 180.degree.), are on the ground, the antennas typically are pointed 180.degree. away from the center of the earth and toward the sky (i.e., up), or within 35-45.degree. of up. In the case where the antennas are on the ground, the alignment of the antennas is not an issue, because all of the antennas are pointed in the direction of the local vertical (or up), and there are a plurality of GPS signals broadcasts from the sky, but none from the ground.
FIG. 5(A) shows an example of a prior art antenna array 40 in which all antennas 10 are aligned with each other, and pointed in the same direction. FIG. 5(B) shows the coverage pattern 42 of the antenna array 40 in body coordinates shown in FIG. 5(A). The lighter shades of the coverage pattern 42 shown in FIG. 5(B) indicate regions with more common antenna 10 coverage.
To ensure that at least one antenna 10 is favorably aligned with the GPS constellation 39 (as shown in FIGS. 4(A)-4(C)), regardless of vehicle orientation, the antennas must be pointed in different directions, as shown in FIGS. 5(C) and 5(E).
To provide a larger GPS signal coverage area, nonaligned antennas 10 are provided, as shown in each of FIGS. 5(C) and 5(E). FIGS. 5(C) and 5(D) show, respectively, an antenna array 44 having 4 nonaligned antennas 10 used for navigation, but not for attitude determination, and the coverage pattern 42 thereof. Nonaligned antennas are antennas pointing in different directions.
Further, FIGS. 5(E) and 5(F) show, respectively, an antenna array 44 having 4 nonaligned antennas 10 used for navigation, but not for attitude determination, and the coverage pattern 42 thereof.
The nonaligned antenna configurations shown in FIGS. 5(C) and 5(E) will also greatly reduce the common field of view between each antenna pair. but would provide greater sky coverage for the antenna array.
The nonaligned antenna configurations shown in FIGS. 5(C) and 5(E) will provide for vehicle navigation solution availability regardless of vehicle orientation, but cannot be used for vehicle attitude determination based on differential carrier phase measurements between each of the nonaligned antennas because the measurements must be adjusted using the present invention to account for the phase contribution due to the circular polarization of the nonaligned antennas. This effect is common mode on aligned antennas shown in FIG. 5(A), and therefore has not been previously accounted for.
In Maintaining GPS Positioning in Steep Turns Using Two Antennas, D. H. Lawrence, et al., ION GPS-95, Palm Springs, Calif., September 1995, a derivation of a correction term for nonaligned antenna boresights was applied to absolute carrier phase positioning, but not to differential carrier phase for attitude determination.
Because the antennas 10 shown in FIGS. 5(C) and 5(E) are not aligned with each other, the antennas 10 shown therein cannot be relied upon, without the present invention, for determining vehicle attitude using GPS carrier signals, as explained below.
As shown in FIG. 6, the GPS signal 11 broadcast by satellite 12 is received by antenna 10 at an offset angle .theta.. The effect of the angle .theta. is removed from the differential carrier phase measurement 20 because antennas 10 are aligned with each other and pointed in the same direction relative to, and are at a great distance from, GPS signal 11, in the conventional system.
A problem with the prior art arises when the offset angle .theta., discussed herein above with respect to FIG. 6, is not the same for each antenna receiving the GPS carrier signal, and, accordingly, is not removed during calculation of the differential carrier phase explained above. FIGS. 5(C) and 5(E) each show examples of nonaligned antennas in which the offset angle .theta. would not be the same for each antenna in the array, and, accordingly, would not be removed during conventional calculation of the differential carrier phase.
A problem with the prior art is that the effect of antenna alignment on differential carrier phase measurements is neglected.
A problem of nonalignment of antennas 10 is that the right hand circularly polarized (RHCP) antenna phase pattern is not common mode and must be modeled to produce correct differential carrier phase measurements.
A further problem in the prior art is that differential carrier phase measurements from the nonaligned antennas cannot be used for attitude determination, accordingly decreasing the useful results from the nonaligned antennas.
Another problem with the conventional method of determining vehicle attitude arises when a vehicle relies upon signals broadcast from a GPS satellite, since a vehicle such as a spacecraft is orbiting the earth very quickly, and the definition of "up" toward the satellites broadcasting the GPS signals changes. FIGS. 4(A) through 4(C), previously explained, illustrate the relative orientation of antennas 10 to GPS satellites 12 (not shown in FIGS. 4(A) through 4(C), but which are in GPS constellation 39).
Still another problem with the prior art is that a single, uniform set of attitude determination algorithms is not provided that supports all of the very different applications of a Low Earth Orbit (LEO) satellite.